Antiderivatives and basic formulas
A cyclist's speedometer gives the speed at every moment. If we only know the speed, we can rebuild how far the cyclist went. Integration is this reverse job: from the rate back to the quantity.
If the derivative of F(x) is f(x), then F(x) is an antiderivative of f(x), and we write ∫ f(x) dx = F(x) + C. The letter C is the constant of integration, because the derivative of a constant is zero, so every antiderivative can be shifted up or down.
Formula: ∫ xⁿ dx = xⁿ⁺¹ / (n + 1) + C, for n ≠ −1
Rule 1: Power rule — Add 1 to the power and divide by the new power.
- Easy: ∫ x⁵ dx = x⁶/6 + C.
- Medium: ∫ 1/x³ dx = ∫ x⁻³ dx = x⁻²/(−2) + C = −1/(2x²) + C.
- Exam-type: ∫ √x dx = ∫ x^(1/2) dx = x^(3/2)/(3/2) + C = (2/3) x^(3/2) + C.
Rule 2: Standard results — Remember the derivative table in reverse.
- Easy: ∫ cos x dx = sin x + C, because d/dx (sin x) = cos x.
- Medium: ∫ (3eˣ + 2/x) dx = 3eˣ + 2 ln|x| + C.
- Exam-type: ∫ 2ˣ dx = 2ˣ / ln 2 + C.
Always write + C for an indefinite integral. Check any answer by differentiating it: you must get back the function inside the integral.
Using the power rule for n = −1. For 1/x the answer is ln|x| + C, not x⁰/0.
Power up by one, divide by the new power, add C.
Practice questions
- 1Find ∫ x⁴ dx.
- 2Find ∫ (3x² + 2) dx.
- 3Find ∫ cos x dx.
- 4Find ∫ (x³ + 1/x) dx.
- 5Find ∫ √x dx.
- 6Find ∫ (eˣ + sin x) dx.
- 7A curve has f′(x) = 6x − 2 and passes through (1, 3). Find f(x).
- 8Find ∫ (x + 1)² dx by first expanding.
Show answers (try them yourself first)
- 1x⁵/5 + C
- 2x³ + 2x + C
- 3sin x + C
- 4x⁴/4 + ln|x| + C
- 5(2/3) x^(3/2) + C
- 6eˣ − cos x + C
- 7f(x) = 3x² − 2x + C; f(1) = 1 + C = 3 gives C = 2, so f(x) = 3x² − 2x + 2
- 8x³/3 + x² + x + C
Properties of indefinite integrals
When a shop bill is the sum of many items, you can add item by item. Integration also works term by term, and a constant multiplier can be taken outside.
Integration is linear. A constant factor comes out of the integral, and the integral of a sum or difference is the sum or difference of the integrals. Two antiderivatives of the same function differ only by a constant.
Formula: ∫ [k f(x) ± g(x)] dx = k ∫ f(x) dx ± ∫ g(x) dx
Rule 1: Constant out, split the sum — Break the integrand into simple terms.
- Easy: ∫ 5x⁴ dx = 5 · x⁵/5 + C = x⁵ + C.
- Medium: ∫ (4x³ − 6x) dx = x⁴ − 3x² + C.
- Exam-type: ∫ (x + 1/x)² dx = ∫ (x² + 2 + 1/x²) dx = x³/3 + 2x − 1/x + C.
Rule 2: Simplify before integrating — Divide or expand so that each term is a power.
- Easy: ∫ (x² + 1)/x dx = ∫ (x + 1/x) dx = x²/2 + ln|x| + C.
- Medium: ∫ tan²x dx = ∫ (sec²x − 1) dx = tan x − x + C.
- Exam-type: ∫ (x⁴ + 1)/x² dx = ∫ (x² + x⁻²) dx = x³/3 − 1/x + C.
The integral of a product is not the product of the integrals. Expand or divide first.
Writing ∫ x · x dx = (x²/2)(x²/2). The correct value is ∫ x² dx = x³/3 + C.
Split, simplify, then integrate each piece.
Practice questions
- 1Find ∫ 5x⁴ dx.
- 2Find ∫ (4x³ − 6x) dx.
- 3Find ∫ (x² + 1)/x dx.
- 4Find ∫ tan²x dx.
- 5Find ∫ (x⁴ + 1)/x² dx.
- 6Is x³/3 + 5 an antiderivative of x²? Give the reason.
- 7Find ∫ (x + 1/x)² dx.
- 8Find ∫ (2x − 3)(x + 1) dx.
Show answers (try them yourself first)
- 1x⁵ + C
- 2x⁴ − 3x² + C
- 3x²/2 + ln|x| + C
- 4tan x − x + C
- 5x³/3 − 1/x + C
- 6Yes, because its derivative is x² and the constant 5 vanishes on differentiation.
- 7x³/3 + 2x − 1/x + C
- 8(2x − 3)(x + 1) = 2x² − x − 3, so the integral is 2x³/3 − x²/2 − 3x + C